Transitions from Digital Communications to Quantum Communications: Concepts and Prospects

By Malek Benslama, Hadj Batatia and Abderraouf Messai

This book addresses the move towards quantum communications, in light of the recent technological developments on photonic crystals and their potential applications in systems.

The authors present the state of the art on extensive quantum communications, the first part of the book being dedicated to the relevant theory; quantum gates such as Deutsch gates, Toffoli gates and Dedekind gates are reviewed with regards to their feasibility as electronic circuits and their implementation in systems, and a comparison is performed in parallel with conventional circuits such as FPGAs and DSPs. The specifics of quantum communication are also revealed through the entanglement and Bell states, and mathematical and physical aspects of quantum optical fibers and photonic crystals are considered in order to optimize the quantum transmissions.

These concepts are linked with relevant, practical examples in the second part of the book, which presents six integrated applications for quantum communications.

• Language: English
• Publisher: Wiley
• Release date: Jul 14, 2016
• ISBN: 9781119330271

Malek Benslama

Malek Benslama is currently Professor at the University of Constantine 1 in Algeria. He is also Doctor of Science with the INP Toulouse in France and a member of the scientific council of the Algerian Space Agency.

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Table of Contents

Cover

Dedication

Title

Copyright

Foreword

Preface

Introduction

List of Acronyms

PART 1: Theory

1 Non-linear Signal Processing

1.1. Distributions

1.2. Variance

1.3. Covariance

1.4. Stationarity

1.5. Bayes inference

1.6. Tensors in signal processing

1.7. Processing the quantum signal

2 Non-Gaussian Processes

2.1. Defining Gaussian processes

2.2. Non-Gaussian processes

2.3. Principal component analysis or Karhunen–Loève transformation

2.4. Sparse Gaussian processes

2.5. Levy process

2.6. Links with quantum communications

3 Sparse Signals and Compressed Sensing

3.1. Sparse Signals

3.2. Compressed sensing

3.3. Compressed sensing and quantum signal

4 The Fourier Transform

4.1. The Classic Fourier Transform

4.2. The Discreet Fourier Transform and the Fast Fourier Transform

4.3. The Fourier Transform and hyper-functions

4.4. Hilbert Transform

4.5. Clifford algebra and the Fourier Transform

4.6. Spinors and quantum signals

5 The Contribution of Arithmetic to Signal Processing

5.1. Gauss sums

5.2. Applications for Gauss sums

6 Riemannian Geometry and Signal Processing

6.1. Context

6.2. Riemannian varieties

6.3. Voronoi cells

6.4. Applications to Voronoi cells

PART 2: Applications

7 MIMO Systems

7.1. Introduction

7.2. A brief history of OFDM

7.3. Multi-carrier technology

7.4. OFDM technique

7.5. Generating OFDM symbols

7.6. Inter-symbol and inter-carrier interference

7.7. Cyclic prefix

7.8. Mathematical model of the OFDM system

7.9. MIMO channels

7.10. The MIMO channel model

7.11. MIMO OFDM channel model

8 Minimizing Interferences in DS–CDMA Systems

8.1. Convolutional encoding

8.2. Structure of convolutive codes

8.3. Polynomial representation

8.4. Graphic representations of convolutive codes

8.5. Decoding algorithms

8.6. Discreet Wavelet Transform (DWT)

8.7. Construction and discreet filtering

8.8. Defining the wavelet function: the place of detail

8.9. Wavelets and filter banks

8.10. Thresholding coefficients

8.11. Simulating results

9 STAP Radar

9.1. Introduction

9.2. Space–time adaptive processing (STAP)

9.3. Structure of the covariance matrix

9.4. Clutter

9.5. Optimal STAP

9.6. Performance measures

9.7. Influence of the radar’s parameters on detection

9.8. Sample matrix inversion algorithm (SMI)

9.9. Conclusion

10 Tracking Radar (Using the Dempster–Shafer Theory)

10.1. Introduction

10.2. Dempster–Shafer theory

10.3. Rules of combination

10.4. Decision rules

10.5. Digital simulation

10.6. Conclusion

11 InSAR Radar

11.1. Introduction

11.2. Coherence

11.3. System model

11.4. Inferometric phase statistics

11.5. Quantitative examples

11.6. Conclusion

12 Telecommunications Networks

12.1. Introduction

12.2. Describing the ad hoc simulated network’s topology

12.3. The different scenarios enacted

12.4. The statistics collected

12.5. Discussion of results

12.6. Part two: network using OLSR for routing

12.7. Conclusion

Conclusion

Bibliography

Index

End User License Agreement

List of Tables

1 Non-linear Signal Processing

Table 1.1. Devices, topics and main discoverers

9 STAP Radar

Table 9.1. Radar system parameters

10 Tracking Radar (Using the Dempster–Shafer Theory)

Table 10.1. Table of combined masses

List of Illustrations

1 Non-linear Signal Processing

Figure 1.1. Generalized Bayesian inference process

2 Non-Gaussian Processes

Figure 2.1. Probability distribution in Q of the linear superposition of two coherent states

Figure 2.2. Probability distribution in P compared to a linear superposition (plain line) and a statistical mix (dotted line) of two coherent states

3 Sparse Signals and Compressed Sensing

Figure 3.1. Sparse discreet-time signal with its DFT

Figure 3.2. Manifestation of sparsity in the frequency domain

Figure 3.3. Block diagram of an iterative reconstruction method. The masking is an appropriate filter with coefficients of 1 and 0 according to the type of sparsity in the original signal

Figure 3.4. Original Gaussian signal with 100 spikes

Figure 3.5. Reconstructed signal and the tally with the coefficients by coefficient of x0 depending on its reconstruction

Figure 3.6. An algorithmic overview of compressed sensing

6 Riemannian Geometry and Signal Processing

Figure 6.1. Superposition of a Voronoi diagram (in red) and its Delaunay triangulation (in black). For a color version of this figure, see www.iste.co.uk/benslama/question.zip

7 MIMO Systems

Figure 7.1. Effect of a fading on serial and parallel systems

Figure 7.2. An OFDM sub-carrier transmitter

Figure 7.3. Subdivision of the bandwidth into N sub-carriers

Figure 7.4. Multi-carrier modulation

Figure 7.5. The overlapped spectrum of an OFDM signal. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

Figure 7.6. OFDM demodulation using correlators

Figure 7.7. Diagram of a MIMO OFDM system

Figure 7.8. Cyclic prefix

Figure 7.9. A basic FFT OFDM transmitter-receiver [GER 05]

Figure 7.10. The continuous time OFDM system interpreted as parallel Gaussian channels

Figure 7.11. A standard approach for a MIMO–OFDM system with four transmitting antennae and four receiving antennae

Figure 7.12. Model of a MIMO–OFDM system

8 Minimizing Interferences in DS–CDMA Systems

Figure 8.1. Convolutive performance coder = 1/n

Figure 8.2. Tree diagram for k = 1, n = 2 and L = 3

Figure 8.3. State diagram for k = 1, n = 2 and L = 3

Figure 8.4. Trellis diagram for k=1, n =2 and L= 3

Figure 8.5. Cubic spline wavelet function and its Fourier transform

Figure 8.6. Direct and inverse fast wavelet transform

Figure 8.7. The TOR for a vector of N=2³ samples

Figure 8.8. Diagram of the threshold principle

Figure 8.9. Hard thresholding

Figure 8.10. Soft thresholding

Figure 8.11. BER according to the SNR with N=63 for a coded channel. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

Figure 8.12. BER according to K with SNR = 0 dB for a coded channel. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

Figure 8.13. BER according to the SNR with N=63 for the developed approach. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

Figure 8.14. Comparison of the BER according to the SNR for the three approaches. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

Figure 8.15. BER according to K for SNR = 0 dB for the developed approach

Figure 8.16. Comparison of BER according to K for the three approaches. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

9 STAP Radar

Figure 9.1. Space–time filter

Figure 9.2. Geometry of the configuration of an airborne side-looking radar

Figure 9.3. Conventional STAP chain

Figure 9.4. General structure of the STAP filter. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

Figure 9.5. STAP data cube

Figure 9.6. Illustration of the clutter model

Figure 9.7. Clutter crests for different values of PRF

Figure 9.8. The optimal STAP filter’s response, at zero degrees of elevation, in the presence of two jammers at -60° and JNR=40 dB, N=12, M=10 and CNR=40 dB. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

Figure 9.9. Improvement factor for the optimal processor DFP. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

Figure 9.10. Improvement factor for the optimal processor DFP with. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

Figure 9.11. Improvement factor for the optimal processor DFP for different spacings: a) , b)

Figure 9.12. Improvement factor for the optimal processor DFP with PRF constant, N = 8, M = 10, d/λ = 0.5:

Figure 9.13. Angle/Doppler spectrum in the presence of two jammers at -40° and 60° with JNR = 45 dB, N=8, M=10, CNR = 20 dB, β = 0.5, β = 1, β = 2. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

Figure 9.14. Improvement factor for the optimal processor DFP with ; a) Bs = 0; b) Bs = 0.1

Figure 9.15. This figure illustrates the filter’s angle/Doppler response using the SMI algorithm. We note that the interferences are cancelled (clutter, jammer) but with a distortion of the secondary lobes by comparison to the optimal STAP. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

Figure 9.16. The filter’s angle/Doppler response using the SMI algorithm target inserted in cell 50, Ft=0.3 and in the presence of two jammers at JNR=40 dB and CNR=20 dB. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

10 Tracking Radar (Using the Dempster–Shafer Theory)

Figure 10.1. Different mass function measures

Figure 10.2. Credibility and plausibility of a set A

Figure 10.3. Representing the evidence of a set A

Figure 10.4. Linking measurements to the corresponding targets

Figure 10.5. Association steps

Figure 10.6. Real trajectories and those estimated using DST for three parallel targets. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

Figure 10.7. Combination rate for the three parallel targets; a) separations between the targets 50 m; b) separations between the targets 100 m; c) separations between the targets 150 m

Figure 10.8. Real trajectories and those estimated using DST for three intersecting targets. For a color version of this figure, see www.iste.co.uk/benslama/quantum.zip

Figure 10.9. Combination rate for three intersecting targets; a) Intersection point N/4; b) Intersection point N/2; c) Intersection point 3N/4

11 InSAR Radar

Figure 11.1. Formation system model INSAR [HAG 70]

Figure 11.2. The inferometric phase’s probability density functin for different values of the correlation coefficient

Figure 11.3. Standard deviation of the interferometric phase compared to the size of the correlation coefficient

Figure 11.4. Standard deviation for the phase compared to the SNR

Figure 11.5. The phase deviation according to the relative change between the two images as a fraction of a resolution element

Figure 11.6. a) The bias phase according to the error phase at the edges of the azimuth bandwidth; b) the phase deviation according to the phase error at the edges of the azimuth bandwidth

Figure 11.7. The phase deviation migration of an incorrect resolution element’s linear gate

Figure 11.8. The phase deviation compared to the non-compensated quadratic gate’s migration expressed in fractions ε

12 Telecommunications Networks

Figure 12.1. Ad hoc topology network introducing the concept of Mutihoming

Figure 12.2. Routing traffic (AODV messages) sent by all the nodes in the network at packets/sec. For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

Figure 12.3. The traffic sent by the source in both scenarios (packets/sec). For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

Figure 12.4. Traffic received by the destination node in both scenarios (packetts/sec). For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

Figure 12.5. AODV packets sent throughout the network. For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

Figure 12.6. Packets received by destination. For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

Figure 12.7. Packets sent by source. For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

Figure 12.8. The traffic load received by router 3 in scenarios 2 and 3. For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

Figure 12.9. Total traffic routing (OLSR message) on the network. For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

Figure 12.10. The traffic sent by the source. For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

Figure 12.11. Traffic received by the destination node. For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

Figure 12.12. Total routing traffic on the network. For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

Figure 12.13. Traffic sent by the source. For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

Figure 12.14. Traffic received by the destination node. For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

Figure 12.15. The traffic load received by router 3 in scenarios 2 and 3. For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

Figure 12.16. Traffic received by router 3 for both protocols. For a color version of this figure, see www.iste.co.uk./benslama/quantum.zip

To my Mother, with my deep gratitude and affection

Series Editor

Guy Pujolle

Transitions from Digital Communications to Quantum Communications

Concepts and Prospects

Malek Benslama

Hadj Batatia

Abderraouf Messai

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First published 2016 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd 2016

The rights of Malek Benslama, Hadj Batatia and Abderraouf Messai to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2016940263

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-925-0

Foreword

Four works dedicated entirely to satellite communications: this is the challenge set by Professor Malek Benslama of the University of Constantine, who understood that a new discipline was in the process of taking shape.

He demonstrated this by organizing the first international symposium on Electromagnetism, Satellites and Cryptography at Jijel in June 2005. The success of congress, surprising for a first-time event, shows that there was a need to gather, in a single place, specialists with skills that are sometimes very removed from one another. The 140 papers accepted concerned systems for electromagnetic systems as well as circuit and antennae engineering and cryptography, which is very often based on pure mathematics. A synergy between these disciplines is necessary to develop the new field of activity that is satellite communication.

The emergence of new disciplines of this type has already taken place before: for electromagnetic compatibility, it was as necessary to know electrical engineering for driven modes and choppers as electromagnetics (radiating modes) and to be able to define specific experimental protocols. Further back in time, we saw the emergence of computing which, at the start, lay in the field of electronics and was able, over time, to become independent.

Professor Benslama has the outlook and open-mindedness indispensable for bringing to fruition the synthesis between the skills that coexist in satellite telecommunications. I have known him for 28 years and for me it is a real pleasure to remember all these years of close acquaintance. There has not been a year in which we have not had an opportunity to see one another. For 15 years he worked on the interaction between acoustic waves and semi-conductors. He specialized in resolving piezoelectric equations (Rayleigh waves, creeping waves, etc.), and at the same time was interested in theoretical physics. A doctoral thesis in engineering and then a state thesis crowned his professional achievements. Notably, his examination committee included Jeannine Henaf, then Chief Engineer for the National Center for Telecommunications studies. He was already interested in telecommunications, but also, with the presence of Michel Planat, responsible for research at CNRS, in the difficult problem of synchronizing oscillators.

It is with Planat that he created the path that would lead to quantum cryptography. He made this transformation over 10 years, thus moving without any apparent difficulty from Maxwell’s equations to Galois groups. He is now therefore one of the people most likely to dominate all those diverse disciplines that form satellite telecommunications.

I wish, with all my friendly admiration, that these four volumes are with a warm welcome from students and teachers